The term "labeled graph" when used without qualification means a graph with each node labeled differently (but arbitrarily), so that all nodes are considered
distinct for purposes of enumeration. The total number of (not necessarily
connected) labeled -node
graphs for ,
2, ... is given by 1, 2, 8, 64, 1024, 32768, ... (OEIS A006125;
illustrated above), and the numbers of connected labeled graphs on -nodes are given by the logarithmic
transform of the preceding sequence, 1, 1, 4, 38, 728, 26704, ... (OEIS A001187;
Sloane and Plouffe 1995, p. 19).
The numbers of graph vertices in all labeled graphs of orders , 2, ... are 1, 4, 24, 256, 5120, 196608, ... (OEIS A095340),
which the numbers of edges are 0, 1, 12, 192, 5120, 245760, ... (OEIS A095351),
the latter of which has closed-form